Translation Invariant Asymptotic Homomorphisms and Extensions of $C^*$-Algebras

Let $A$ and $B$ be $C^*$-algebras, let $A$ be separable, and let $B$ be $\sigma$-unital and stable. We introduce the notion of translation invariance for asymptotic homomorphisms from $SA=C_0(\mathbb{R})\otimes A$ to $B$ and show that the Connes–Higson construction applied to any extension of $A$ by $B$ is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of $A$ by $B$ from a translation invariant asymptotic homomorphism. This leads to our main result that the homotopy classes of extensions coincide with the homotopy classes of translation invariant asymptotic homomorphisms.